The equation formed by multiplying each root of `ax^(2) + bx+ c = 0" by "2 " is "x^(2) = 36x + 24 =0`

Options

A. 2
B. 3
C. 5
D. 8

Explanation

To solve the problem, we need to find the roots of the given quadratic equation and then determine the difference between those roots. The equation provided is:

\[ ax^2 + bx + c = 0 \]

However, the equation formed by multiplying each root by 2 is given as:

\[ x^2 - 36x - 24 = 0 \]

### Step 1: Identify the new equation
The new equation is:

\[ x^2 - 36x - 24 = 0 \]

### Step 2: Find the roots of the new equation
To find the roots of the equation \( x^2 - 36x - 24 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \( a = 1 \), \( b = -36 \), and \( c = -24 \).

### Step 3: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = (-36)^2 - 4(1)(-24) = 1296 + 96 = 1392 \]

### Step 4: Calculate the roots
Now we can substitute the values into the quadratic formula:

\[ x = \frac{36 \pm \sqrt{1392}}{2} \]

Calculating \( \sqrt{1392} \):

\[ \sqrt{1392} = 12\sqrt{13} \]

Now substituting back:

\[ x = \frac{36 \pm 12\sqrt{13}}{2} \]

This simplifies to:

\[ x = 18 \pm 6\sqrt{13} \]

Thus, the roots are:

\[ x_1 = 18 + 6\sqrt{13}, \quad x_2 = 18 - 6\sqrt{13} \]

### Step 5: Find the difference of the roots
The difference of the roots \( x_1 \) and \( x_2 \) is:

\[ \text{Difference} = x_1 - x_2 = (18 + 6\sqrt{13}) - (18 - 6\sqrt{13}) = 12\sqrt{13} \]

### Final Answer
The difference between the roots is:

\[ 12\sqrt{13} \]